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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p><dfn class="terminology">Case 1</dfn> <span class="process-math">\(u(x, y)\)</span> is a function of <span class="process-math">\(x\)</span> onlyWe want to find the necessary condition that <span class="process-math">\(u(x, y)\)</span> is a function of <span class="process-math">\(x\)</span> only. From (<a href="" class="xref" data-knowl="./knowl/eq2_30.html" title="Equation 2.6.7">(2.6.7)</a>), we have</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_30.html">
\begin{equation*}
-N \frac{\textrm{d} u}{\textrm{d} x}+\left( \frac{\partial M}{\partial y}-\frac{\partial N}{\partial x} \right) u=0~\rightarrow~\frac{1}{u} \frac{\textrm{d} u}{\textrm{d} x}=\frac{ \frac{\partial M}{\partial y}-\frac{\partial N}{\partial x} }{N}.
\end{equation*}
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<p class="continuation">The left hand side of the equation is a function of <span class="process-math">\(x\)</span> only, thus it is necessary that</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_30.html">
\begin{equation*}
\frac{ \frac{\partial M}{\partial y}-\frac{\partial N}{\partial x} }{N}
\end{equation*}
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<p class="continuation">is also a function of <span class="process-math">\(x\)</span> only.</p>
<span class="incontext"><a href="sec2_6.html#p-52" class="internal">in-context</a></span>
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